Construct Non-Hierarchical P/NBD Model for Online Retail Transaction Data

Author

Mick Cooney

Published

July 14, 2023

In this workbook we construct our first hierarchical P/NBD models on the synthetic data with the longer timeframe.

1 Load and Construct Datasets

We start by modelling the P/NBD model using our synthetic datasets before we try to model real-life data.

Show code
use_fit_start_date <- as.Date("2009-12-01")
use_fit_end_date   <- as.Date("2010-12-01")

use_valid_start_date <- as.Date("2010-12-01")
use_valid_end_date   <- as.Date("2011-12-31")

1.1 Load Online Retail Data

We now want to load the short time-frame synthetic data.

Show code
customer_cohortdata_tbl <- read_rds("data/onlineretail_cohort_tbl.rds")
customer_cohortdata_tbl |> glimpse()
Rows: 5,852
Columns: 5
$ customer_id     <chr> "12346", "12347", "12348", "12349", "12350", "12351", …
$ cohort_qtr      <chr> "2010 Q1", "2010 Q4", "2010 Q3", "2010 Q2", "2011 Q1",…
$ cohort_ym       <chr> "2010 03", "2010 10", "2010 09", "2010 04", "2011 02",…
$ first_tnx_date  <date> 2010-03-02, 2010-10-31, 2010-09-27, 2010-04-29, 2011-…
$ total_tnx_count <int> 3, 8, 5, 3, 1, 1, 9, 2, 1, 2, 6, 2, 5, 10, 6, 4, 10, 2…
Show code
customer_transactions_tbl <- read_rds("data/onlineretail_invoice_cleaned_tbl.rds")
customer_transactions_tbl |> glimpse()
Rows: 36,594
Columns: 4
$ invoice_date  <date> 2009-12-01, 2009-12-01, 2009-12-01, 2009-12-01, 2009-12…
$ customer_id   <chr> "12490", "12533", "12682", "12758", "12836", "12913", "1…
$ invoice_id    <chr> "489557", "489526", "489439", "489599", "489593", "48954…
$ invoice_spend <dbl> 531.94, 821.92, 372.30, 2454.68, 423.87, 537.96, 261.00,…

1.2 Load Derived Data

Show code
customer_summarystats_tbl <- read_rds("data/onlineretail_customer_summarystats_tbl.rds")

obs_fitdata_tbl   <- read_rds("data/onlineretail_obs_fitdata_tbl.rds")
obs_validdata_tbl <- read_rds("data/onlineretail_obs_validdata_tbl.rds")

customer_fit_stats_tbl <- obs_fitdata_tbl |>
  rename(x = tnx_count)

2 Fit First P/NBD Model

We now construct our Stan model and prepare to fit it with our synthetic dataset.

Before we start on that, we set a few parameters for the workbook to organise our Stan code.

Show code
stan_modeldir <- "stan_models"
stan_codedir  <-   "stan_code"

We also want to set a number of overall parameters for this workbook

To start the fit data, we want to use the 1,000 customers. We also need to calculate the summary statistics for the validation period.

2.1 Compile and Fit Stan Model

We now compile this model using CmdStanR.

Show code
pnbd_fixed_stanmodel <- cmdstan_model(
  "stan_code/pnbd_fixed.stan",
  include_paths =   stan_codedir,
  pedantic      =           TRUE,
  dir           =  stan_modeldir
  )

We then use this compiled model with our data to produce a fit of the data.

Show code
stan_modelname <- "pnbd_onlineretail_fixed1"
stanfit_seed   <- stanfit_seed + 1
stanfit_prefix <- str_c("fit_", stan_modelname) 

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")

stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    lambda_mn = 0.25,
    lambda_cv = 1.00,
    
    mu_mn     = 0.10,
    mu_cv     = 1.00,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_onlineretail_fixed1_stanfit <- pnbd_fixed_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )
  
  pnbd_onlineretail_fixed1_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  pnbd_onlineretail_fixed1_stanfit <- read_rds(stanfit_object_file)
}

pnbd_onlineretail_fixed1_stanfit$summary()
# A tibble: 12,718 × 10
   variable       mean   median      sd     mad       q5      q95  rhat ess_bulk
   <chr>         <num>    <num>   <num>   <num>    <num>    <num> <num>    <num>
 1 lp__       -6.82e+4 -6.82e+4 71.9    71.7    -6.83e+4 -6.81e+4  1.01     704.
 2 lambda[1]   6.43e-2  5.20e-2  0.0497  0.0409  1.04e-2  1.59e-1  1.00    1962.
 3 lambda[2]   1.41e-1  9.80e-2  0.148   0.0972  6.83e-3  4.24e-1  1.00    1817.
 4 lambda[3]   1.33e-1  8.00e-2  0.160   0.0884  4.79e-3  4.41e-1  1.00    1351.
 5 lambda[4]   5.78e-2  4.77e-2  0.0408  0.0350  1.11e-2  1.37e-1  1.00    1626.
 6 lambda[5]   2.47e-1  1.61e-1  0.271   0.174   1.03e-2  7.39e-1  1.00    1200.
 7 lambda[6]   3.09e-1  2.59e-1  0.216   0.185   5.64e-2  7.29e-1  1.00    2131.
 8 lambda[7]   1.42e-1  9.21e-2  0.155   0.0979  7.57e-3  4.44e-1  1.00    1678.
 9 lambda[8]   1.35e-1  7.92e-2  0.161   0.0886  4.57e-3  4.50e-1  1.00    1374.
10 lambda[9]   2.70e-1  2.45e-1  0.154   0.140   7.44e-2  5.52e-1  1.00    1920.
# ℹ 12,708 more rows
# ℹ 1 more variable: ess_tail <num>

We have some basic HMC-based validity statistics we can check.

Show code
pnbd_onlineretail_fixed1_stanfit$cmdstan_diagnose()
Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_fixed1-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_fixed1-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_fixed1-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_fixed1-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

2.2 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Show code
parameter_subset <- c(
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_onlineretail_fixed1_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We also check \(N_{eff}\) as a quick diagnostic of the fit.

Show code
pnbd_onlineretail_fixed1_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  as.numeric() |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

2.3 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

Show code
pnbd_onlineretail_fixed1_assess_data_lst <- run_model_assessment(
  model_stanfit    = pnbd_onlineretail_fixed1_stanfit,
  insample_tbl     = customer_fit_stats_tbl,
  outsample_tbl    = customer_valid_stats_tbl,
  fit_label        = "pnbd_onlineretail_fixed1",
  fit_end_dttm     = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm   = use_valid_end_date   |> as.POSIXct(),
  sim_seed         = 10
  )

pnbd_onlineretail_fixed1_assess_data_lst |> glimpse()
List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_onlineretail_fixed1_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_onlineretail_fixed1_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_onlineretail_fixed1_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_onlineretail_fixed1_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_onlineretail_fixed1_assess_valid_simstats_tbl.rds"

2.3.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Show code
simdata_tbl <- pnbd_onlineretail_fixed1_assess_data_lst |>
  use_series(model_fit_simstats_filepath) |>
  read_rds()

insample_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = obs_fitdata_tbl,
  simdata_tbl = simdata_tbl
  )

insample_plots_lst$multi_plot |> print()

Show code
insample_plots_lst$total_plot |> print()

Show code
insample_plots_lst$quant_plot |> print()

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

2.3.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Show code
simdata_tbl <- pnbd_onlineretail_fixed1_assess_data_lst |>
  use_series(model_valid_simstats_filepath) |>
  read_rds()

outsample_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = obs_validdata_tbl,
  simdata_tbl = simdata_tbl
  )

outsample_plots_lst$multi_plot |> print()

Show code
outsample_plots_lst$total_plot |> print()

Show code
outsample_plots_lst$quant_plot |> print()

As for our short time frame data, overall our model is working well.

           used  (Mb) gc trigger   (Mb)  max used   (Mb)
Ncells  3771669 201.5    6554501  350.1   6554501  350.1
Vcells 83250000 635.2  256750661 1958.9 256749976 1958.9

3 Fit Alternate Prior Model.

We want to try an alternate prior model with a smaller co-efficient of variation to see what impact it has on our procedures.

Show code
stan_modelname <- "pnbd_onlineretail_fixed2"
stanfit_seed   <- stanfit_seed + 1
stanfit_prefix <- str_c("fit_", stan_modelname) 

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")

stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    lambda_mn = 0.25,
    lambda_cv = 0.50,
    
    mu_mn     = 0.10,
    mu_cv     = 0.50,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_onlineretail_fixed2_stanfit <- pnbd_fixed_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )
  
  pnbd_onlineretail_fixed2_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  pnbd_onlineretail_fixed2_stanfit <- read_rds(stanfit_object_file)
}

pnbd_onlineretail_fixed2_stanfit$summary()
# A tibble: 12,718 × 10
   variable       mean   median      sd     mad       q5      q95  rhat ess_bulk
   <chr>         <num>    <num>   <num>   <num>    <num>    <num> <num>    <num>
 1 lp__       -1.44e+5 -1.44e+5 67.2    66.7    -1.44e+5 -1.44e+5  1.00     719.
 2 lambda[1]   1.32e-1  1.21e-1  0.0625  0.0582  4.95e-2  2.51e-1  1.00    3089.
 3 lambda[2]   2.11e-1  1.93e-1  0.108   0.0999  7.49e-2  4.12e-1  1.00    2902.
 4 lambda[3]   2.07e-1  1.86e-1  0.108   0.0955  7.00e-2  4.12e-1  1.00    3208.
 5 lambda[4]   1.09e-1  1.02e-1  0.0475  0.0431  4.30e-2  1.99e-1  1.00    3086.
 6 lambda[5]   2.49e-1  2.26e-1  0.129   0.119   7.89e-2  4.93e-1  1.00    4277.
 7 lambda[6]   2.69e-1  2.50e-1  0.119   0.113   1.09e-1  4.97e-1  1.00    3642.
 8 lambda[7]   2.09e-1  1.90e-1  0.109   0.103   6.30e-2  4.09e-1  1.00    3500.
 9 lambda[8]   2.12e-1  1.96e-1  0.109   0.101   6.60e-2  4.20e-1  1.01    3648.
10 lambda[9]   2.60e-1  2.44e-1  0.107   0.103   1.12e-1  4.61e-1  1.01    3962.
# ℹ 12,708 more rows
# ℹ 1 more variable: ess_tail <num>

We have some basic HMC-based validity statistics we can check.

Show code
pnbd_onlineretail_fixed2_stanfit$cmdstan_diagnose()
Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_fixed2-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_fixed2-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_fixed2-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_fixed2-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

3.1 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Show code
parameter_subset <- c(
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_onlineretail_fixed2_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We want to check the \(N_{eff}\) statistics also.

Show code
pnbd_onlineretail_fixed2_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  as.numeric() |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

3.2 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

Show code
pnbd_onlineretail_fixed2_assess_data_lst <- run_model_assessment(
  model_stanfit    = pnbd_onlineretail_fixed2_stanfit,
  insample_tbl     = customer_fit_stats_tbl,
  outsample_tbl    = customer_valid_stats_tbl,
  fit_label        = "pnbd_onlineretail_fixed2",
  fit_end_dttm     = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm   = use_valid_end_date   |> as.POSIXct(),
  sim_seed         = 20
  )

pnbd_onlineretail_fixed2_assess_data_lst |> glimpse()
List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_onlineretail_fixed2_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_onlineretail_fixed2_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_onlineretail_fixed2_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_onlineretail_fixed2_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_onlineretail_fixed2_assess_valid_simstats_tbl.rds"

3.2.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Show code
simdata_tbl <- pnbd_onlineretail_fixed2_assess_data_lst |>
  use_series(model_fit_simstats_filepath) |>
  read_rds()

insample_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = obs_fitdata_tbl,
  simdata_tbl = simdata_tbl
  )

insample_plots_lst$multi_plot |> print()

Show code
insample_plots_lst$total_plot |> print()

Show code
insample_plots_lst$quant_plot |> print()

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

3.2.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Show code
simdata_tbl <- pnbd_onlineretail_fixed2_assess_data_lst |>
  use_series(model_valid_simstats_filepath) |>
  read_rds()

outsample_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = obs_validdata_tbl,
  simdata_tbl = simdata_tbl
  )

outsample_plots_lst$multi_plot |> print()

Show code
outsample_plots_lst$total_plot |> print()

Show code
outsample_plots_lst$quant_plot |> print()

            used   (Mb) gc trigger   (Mb)  max used   (Mb)
Ncells   3797211  202.8    6554501  350.1   6554501  350.1
Vcells 134254213 1024.3  308180793 2351.3 308179829 2351.3

4 Fit Tight-Lifetime Model

We now want to try a model where we use priors with a tighter coefficient of variation for lifetime but keep the CoV for transaction frequency.

Show code
stan_modelname <- "pnbd_onlineretail_fixed3"
stanfit_seed   <- stanfit_seed + 1
stanfit_prefix <- str_c("fit_", stan_modelname) 

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    lambda_mn = 0.25,
    lambda_cv = 1.00,
    
    mu_mn     = 0.10,
    mu_cv     = 0.50,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_onlineretail_fixed3_stanfit <- pnbd_fixed_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )
  
  pnbd_onlineretail_fixed3_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  pnbd_onlineretail_fixed3_stanfit <- read_rds(stanfit_object_file)
}

pnbd_onlineretail_fixed3_stanfit$summary()
# A tibble: 12,718 × 10
   variable       mean   median      sd     mad       q5      q95  rhat ess_bulk
   <chr>         <num>    <num>   <num>   <num>    <num>    <num> <num>    <num>
 1 lp__       -1.12e+5 -1.12e+5 69.6    69.7    -1.12e+5 -1.12e+5  1.00     780.
 2 lambda[1]   7.15e-2  5.77e-2  0.0542  0.0429  1.15e-2  1.79e-1  1.00    2655.
 3 lambda[2]   1.49e-1  9.94e-2  0.162   0.105   7.06e-3  4.46e-1  1.00    1844.
 4 lambda[3]   1.34e-1  8.13e-2  0.158   0.0858  6.35e-3  4.43e-1  1.00    2267.
 5 lambda[4]   5.92e-2  5.00e-2  0.0415  0.0377  9.93e-3  1.39e-1  1.00    2612.
 6 lambda[5]   2.36e-1  1.69e-1  0.228   0.166   1.38e-2  6.98e-1  1.00    2197.
 7 lambda[6]   3.00e-1  2.55e-1  0.208   0.187   5.53e-2  7.02e-1  1.00    2496.
 8 lambda[7]   1.44e-1  9.25e-2  0.160   0.0985  6.70e-3  4.23e-1  1.00    2262.
 9 lambda[8]   1.34e-1  8.35e-2  0.154   0.0923  5.13e-3  4.34e-1  1.00    2776.
10 lambda[9]   2.69e-1  2.38e-1  0.158   0.144   6.92e-2  5.62e-1  1.00    2091.
# ℹ 12,708 more rows
# ℹ 1 more variable: ess_tail <num>

We have some basic HMC-based validity statistics we can check.

Show code
pnbd_onlineretail_fixed3_stanfit$cmdstan_diagnose()
Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_fixed3-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_fixed3-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_fixed3-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_fixed3-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

4.1 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Show code
parameter_subset <- c(
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_onlineretail_fixed3_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We want to check the \(N_{eff}\) statistics also.

Show code
pnbd_onlineretail_fixed3_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  as.numeric() |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

4.2 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

Show code
pnbd_onlineretail_fixed3_assess_data_lst <- run_model_assessment(
  model_stanfit    = pnbd_onlineretail_fixed3_stanfit,
  insample_tbl     = customer_fit_stats_tbl,
  outsample_tbl    = customer_valid_stats_tbl,
  fit_label        = "pnbd_onlineretail_fixed3",
  fit_end_dttm     = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm   = use_valid_end_date   |> as.POSIXct(),
  sim_seed         = 30
  )

pnbd_onlineretail_fixed3_assess_data_lst |> glimpse()
List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_onlineretail_fixed3_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_onlineretail_fixed3_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_onlineretail_fixed3_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_onlineretail_fixed3_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_onlineretail_fixed3_assess_valid_simstats_tbl.rds"

4.2.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Show code
simdata_tbl <- pnbd_onlineretail_fixed3_assess_data_lst |>
  use_series(model_fit_simstats_filepath) |>
  read_rds()

insample_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = obs_fitdata_tbl,
  simdata_tbl = simdata_tbl
  )

insample_plots_lst$multi_plot |> print()

Show code
insample_plots_lst$total_plot |> print()

Show code
insample_plots_lst$quant_plot |> print()

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

4.2.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Show code
simdata_tbl <- pnbd_onlineretail_fixed3_assess_data_lst |>
  use_series(model_valid_simstats_filepath) |>
  read_rds()

outsample_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = obs_validdata_tbl,
  simdata_tbl = simdata_tbl
  )

outsample_plots_lst$multi_plot |> print()

Show code
outsample_plots_lst$total_plot |> print()

Show code
outsample_plots_lst$quant_plot |> print()

            used   (Mb) gc trigger   (Mb)  max used   (Mb)
Ncells   3821935  204.2    6554501  350.1   6554501  350.1
Vcells 185256404 1413.4  443956341 3387.2 369895832 2822.1

5 Fit Narrow-Short-Lifetime Model

We now want to try a model where we use priors with a tighter coefficient of variation for lifetime but keep the CoV for transaction frequency.

Show code
stan_modelname <- "pnbd_onlineretail_fixed4"
stanfit_seed   <- stanfit_seed + 1
stanfit_prefix <- str_c("fit_", stan_modelname) 

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    lambda_mn = 0.25,
    lambda_cv = 1.00,
    
    mu_mn     = 0.20,
    mu_cv     = 0.30,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_onlineretail_fixed4_stanfit <- pnbd_fixed_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )
  
  pnbd_onlineretail_fixed4_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  pnbd_onlineretail_fixed4_stanfit <- read_rds(stanfit_object_file)
}

pnbd_onlineretail_fixed4_stanfit$summary()
# A tibble: 12,718 × 10
   variable       mean   median      sd     mad       q5      q95  rhat ess_bulk
   <chr>         <num>    <num>   <num>   <num>    <num>    <num> <num>    <num>
 1 lp__       -1.85e+5 -1.85e+5 73.2    72.6    -1.85e+5 -1.85e+5 1.01      508.
 2 lambda[1]   8.12e-2  6.75e-2  0.0574  0.0499  1.64e-2  1.92e-1 1.00     2576.
 3 lambda[2]   1.62e-1  1.09e-1  0.182   0.115   4.14e-3  5.07e-1 1.00     1526.
 4 lambda[3]   1.61e-1  1.02e-1  0.183   0.109   6.79e-3  5.06e-1 1.00     2034.
 5 lambda[4]   6.06e-2  5.05e-2  0.0427  0.0379  1.06e-2  1.45e-1 1.00     2772.
 6 lambda[5]   2.37e-1  1.65e-1  0.231   0.170   1.36e-2  7.13e-1 1.00     2523.
 7 lambda[6]   3.02e-1  2.53e-1  0.205   0.182   5.74e-2  6.85e-1 0.999    2449.
 8 lambda[7]   1.61e-1  1.09e-1  0.172   0.114   7.40e-3  5.07e-1 1.00     1919.
 9 lambda[8]   1.65e-1  1.05e-1  0.182   0.114   7.23e-3  5.39e-1 1.00     2233.
10 lambda[9]   2.70e-1  2.42e-1  0.152   0.135   7.81e-2  5.74e-1 1.00     2380.
# ℹ 12,708 more rows
# ℹ 1 more variable: ess_tail <num>

We have some basic HMC-based validity statistics we can check.

Show code
pnbd_onlineretail_fixed4_stanfit$cmdstan_diagnose()
Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_fixed4-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_fixed4-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_fixed4-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_fixed4-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

5.1 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Show code
parameter_subset <- c(
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_onlineretail_fixed4_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We want to check the \(N_{eff}\) statistics also.

Show code
pnbd_onlineretail_fixed4_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  as.numeric() |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

5.2 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

Show code
pnbd_onlineretail_fixed4_assess_data_lst <- run_model_assessment(
  model_stanfit    = pnbd_onlineretail_fixed4_stanfit,
  insample_tbl     = customer_fit_stats_tbl,
  outsample_tbl    = customer_valid_stats_tbl,
  fit_label        = "pnbd_onlineretail_fixed4",
  fit_end_dttm     = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm   = use_valid_end_date   |> as.POSIXct(),
  sim_seed         = 40
  )

pnbd_onlineretail_fixed4_assess_data_lst |> glimpse()
List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_onlineretail_fixed4_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_onlineretail_fixed4_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_onlineretail_fixed4_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_onlineretail_fixed4_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_onlineretail_fixed4_assess_valid_simstats_tbl.rds"

5.2.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Show code
simdata_tbl <- pnbd_onlineretail_fixed4_assess_data_lst |>
  use_series(model_fit_simstats_filepath) |>
  read_rds()

insample_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = obs_fitdata_tbl,
  simdata_tbl = simdata_tbl
  )

insample_plots_lst$multi_plot |> print()

Show code
insample_plots_lst$total_plot |> print()

Show code
insample_plots_lst$quant_plot |> print()

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

5.2.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Show code
simdata_tbl <- pnbd_onlineretail_fixed4_assess_data_lst |>
  use_series(model_valid_simstats_filepath) |>
  read_rds()

outsample_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = obs_validdata_tbl,
  simdata_tbl = simdata_tbl
  )

outsample_plots_lst$multi_plot |> print()

Show code
outsample_plots_lst$total_plot |> print()

Show code
outsample_plots_lst$quant_plot |> print()

            used   (Mb) gc trigger   (Mb)  max used   (Mb)
Ncells   3846672  205.5    6554504  350.1   6554504  350.1
Vcells 236258635 1802.6  532827609 4065.2 443955954 3387.2

6 Compare Model Outputs

We have looked at each of the models individually, but it is also worth looking at each of the models as a group.

Show code
calculate_simulation_statistics <- function(file_rds) {
  simdata_tbl <- read_rds(file_rds)
  
  multicount_cust_tbl <- simdata_tbl |>
    filter(sim_tnx_count > 0) |>
    count(draw_id, name = "multicust_count")
  
  totaltnx_data_tbl <- simdata_tbl |>
    count(draw_id, wt = sim_tnx_count, name = "simtnx_count")
  
  simstats_tbl <- multicount_cust_tbl |>
    inner_join(totaltnx_data_tbl, by = "draw_id")
  
  return(simstats_tbl)
}
Show code
obs_fit_customer_count <- obs_fitdata_tbl |>
  filter(tnx_count > 0) |>
  nrow()

obs_valid_customer_count <- obs_validdata_tbl |>
  filter(tnx_count > 0) |>
  nrow()

obs_fit_total_count <- obs_fitdata_tbl |>
  pull(tnx_count) |>
  sum()

obs_valid_total_count <- obs_validdata_tbl |>
  pull(tnx_count) |>
  sum()

obs_stats_tbl <- tribble(
  ~assess_type, ~name,               ~obs_value,
  "fit",        "multicust_count",   obs_fit_customer_count,
  "fit",        "simtnx_count",      obs_fit_total_count,
  "valid",      "multicust_count",   obs_valid_customer_count,
  "valid",      "simtnx_count",      obs_valid_total_count
  )

model_assess_tbl <- dir_ls("data", regexp = "pnbd_onlineretail_fixed.*_assess_.*simstats") |>
  enframe(name = NULL, value = "file_path") |>
  filter(str_detect(file_path, "_assess_model_", negate = TRUE)) |>
  mutate(
    model_label = str_replace(file_path, "data/pnbd_onlineretail_(.*?)_assess_.*", "\\1"),
    assess_type = if_else(str_detect(file_path, "_assess_fit_"), "fit", "valid"),
    
    sim_data = map(
      file_path, calculate_simulation_statistics,
      
      .progress = "calculate_simulation_statistics"
      )
    )

model_assess_tbl |> glimpse()
Rows: 8
Columns: 4
$ file_path   <fs::path> "data/pnbd_onlineretail_fixed1_assess_fit_simstats_tb…
$ model_label <chr> "fixed1", "fixed1", "fixed2", "fixed2", "fixed3", "fixed3"…
$ assess_type <chr> "fit", "valid", "fit", "valid", "fit", "valid", "fit", "va…
$ sim_data    <list> [<tbl_df[2000 x 3]>], [<tbl_df[2000 x 3]>], [<tbl_df[2000…

We have now constructed the simulation summary statistics and now reshape our data to aid in our model assessment.

Show code
model_assess_summstat_tbl <- model_assess_tbl |>
  select(model_label, assess_type, sim_data) |>
  unnest(sim_data) |>
  pivot_longer(
    cols = !c(model_label, assess_type, draw_id)
    ) |>
  group_by(model_label, assess_type, name) |>
  summarise(
    .groups = "drop",
    
    mean_val = mean(value),
    p10 = quantile(value, 0.10),
    p25 = quantile(value, 0.25),
    p50 = quantile(value, 0.50),
    p75 = quantile(value, 0.75),
    p90 = quantile(value, 0.90)
    )

model_assess_summstat_tbl |> glimpse()
Rows: 16
Columns: 9
$ model_label <chr> "fixed1", "fixed1", "fixed1", "fixed1", "fixed2", "fixed2"…
$ assess_type <chr> "fit", "fit", "valid", "valid", "fit", "fit", "valid", "va…
$ name        <chr> "multicust_count", "simtnx_count", "multicust_count", "sim…
$ mean_val    <dbl> 2594.3130, 12441.8265, 1886.9515, 13200.3765, 2745.4525, 1…
$ p10         <dbl> 2556.0, 12113.9, 1853.0, 12790.9, 2707.0, 9987.9, 1417.0, …
$ p25         <dbl> 2574.0, 12268.0, 1870.0, 12980.0, 2725.0, 10131.0, 1434.0,…
$ p50         <dbl> 2595.0, 12442.0, 1887.0, 13206.5, 2746.0, 10282.0, 1450.0,…
$ p75         <dbl> 2614.00, 12608.25, 1905.00, 13414.50, 2766.00, 10422.25, 1…
$ p90         <dbl> 2632.0, 12771.0, 1919.1, 13604.0, 2784.0, 10562.0, 1483.0,…

We now use this data to construct model comparison plots for the different models we have fit.

Show code
#! echo: TRUE

ggplot(model_assess_summstat_tbl) +
  geom_errorbar(
    aes(x = model_label, ymin = p10, ymax = p90), width = 0
    ) +
  geom_errorbar(
    aes(x = model_label, ymin = p25, ymax = p75), width = 0, linewidth = 3
    ) +
  geom_hline(
    aes(yintercept = obs_value),
    data = obs_stats_tbl, colour = "red"
    ) +
  scale_y_continuous(labels = label_comma()) +
  expand_limits(y = 0) +
  facet_wrap(
    vars(assess_type, name), scale = "free_y"
    ) +
  labs(
    x = "Model",
    y = "Count",
    title = "Comparison Plot for the Different Models"
    ) +
  theme(
    axis.text.x = element_text(angle = 20, vjust = 0.5, size = 8)
    )

6.1 Write Assessment Data to Disk

We now want to save the assessment data to disk.

Show code
model_assess_tbl |> write_rds("data/assess_data_pnbd_onlineretail_fixed_tbl.rds")

7 R Environment

Show code
options(width = 120L)
sessioninfo::session_info()
─ Session info ───────────────────────────────────────────────────────────────────────────────────────────────────────
 setting  value
 version  R version 4.2.3 (2023-03-15)
 os       Ubuntu 22.04.2 LTS
 system   x86_64, linux-gnu
 ui       X11
 language (EN)
 collate  en_US.UTF-8
 ctype    en_US.UTF-8
 tz       Europe/Dublin
 date     2023-07-14
 pandoc   2.19.2 @ /usr/local/bin/ (via rmarkdown)

─ Packages ───────────────────────────────────────────────────────────────────────────────────────────────────────────
 package        * version   date (UTC) lib source
 abind            1.4-5     2016-07-21 [1] RSPM (R 4.2.0)
 arrayhelpers     1.1-0     2020-02-04 [1] RSPM (R 4.2.0)
 backports        1.4.1     2021-12-13 [1] RSPM (R 4.2.0)
 base64enc        0.1-3     2015-07-28 [1] RSPM (R 4.2.0)
 bayesplot      * 1.10.0    2022-11-16 [1] RSPM (R 4.2.0)
 bit              4.0.5     2022-11-15 [1] RSPM (R 4.2.0)
 bit64            4.0.5     2020-08-30 [1] RSPM (R 4.2.0)
 boot             1.3-28.1  2022-11-22 [2] CRAN (R 4.2.3)
 bridgesampling   1.1-2     2021-04-16 [1] RSPM (R 4.2.0)
 brms           * 2.19.0    2023-03-14 [1] RSPM (R 4.2.0)
 Brobdingnag      1.2-9     2022-10-19 [1] RSPM (R 4.2.0)
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 callr            3.7.3     2022-11-02 [1] RSPM (R 4.2.0)
 checkmate        2.1.0     2022-04-21 [1] RSPM (R 4.2.0)
 cli              3.6.1     2023-03-23 [1] RSPM (R 4.2.0)
 cmdstanr       * 0.5.3     2023-06-29 [1] Github (stan-dev/cmdstanr@22b391e)
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 codetools        0.2-19    2023-02-01 [2] CRAN (R 4.2.3)
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 colourpicker     1.2.0     2022-10-28 [1] RSPM (R 4.2.0)
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 lubridate      * 1.9.2     2023-02-10 [1] RSPM (R 4.2.0)
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 minqa            1.2.5     2022-10-19 [1] RSPM (R 4.2.0)
 munsell          0.5.0     2018-06-12 [1] RSPM (R 4.2.0)
 mvtnorm          1.1-3     2021-10-08 [1] RSPM (R 4.2.0)
 nlme             3.1-162   2023-01-31 [2] CRAN (R 4.2.3)
 nloptr           2.0.3     2022-05-26 [1] RSPM (R 4.2.0)
 parallelly       1.35.0    2023-03-23 [1] RSPM (R 4.2.0)
 pillar           1.9.0     2023-03-22 [1] RSPM (R 4.2.0)
 pkgbuild         1.4.0     2022-11-27 [1] RSPM (R 4.2.0)
 pkgconfig        2.0.3     2019-09-22 [1] RSPM (R 4.2.0)
 plyr             1.8.8     2022-11-11 [1] RSPM (R 4.2.0)
 posterior      * 1.4.1     2023-03-14 [1] RSPM (R 4.2.0)
 prettyunits      1.1.1     2020-01-24 [1] RSPM (R 4.2.0)
 processx         3.8.1     2023-04-18 [1] RSPM (R 4.2.0)
 projpred         2.5.0     2023-04-05 [1] RSPM (R 4.2.0)
 promises         1.2.0.1   2021-02-11 [1] RSPM (R 4.2.0)
 ps               1.7.5     2023-04-18 [1] RSPM (R 4.2.0)
 purrr          * 1.0.1     2023-01-10 [1] RSPM (R 4.2.0)
 quadprog         1.5-8     2019-11-20 [1] RSPM (R 4.2.0)
 R6               2.5.1     2021-08-19 [1] RSPM (R 4.2.0)
 Rcpp           * 1.0.10    2023-01-22 [1] RSPM (R 4.2.0)
 RcppParallel     5.1.7     2023-02-27 [1] RSPM (R 4.2.0)
 readr          * 2.1.4     2023-02-10 [1] RSPM (R 4.2.0)
 reshape2         1.4.4     2020-04-09 [1] RSPM (R 4.2.0)
 rlang          * 1.1.0     2023-03-14 [1] RSPM (R 4.2.0)
 rmarkdown        2.21      2023-03-26 [1] RSPM (R 4.2.0)
 rstan            2.21.8    2023-01-17 [1] RSPM (R 4.2.0)
 rstantools       2.3.1     2023-03-30 [1] RSPM (R 4.2.0)
 rsyslog        * 1.0.2     2021-06-04 [1] RSPM (R 4.2.0)
 scales         * 1.2.1     2022-08-20 [1] RSPM (R 4.2.0)
 sessioninfo      1.2.2     2021-12-06 [1] RSPM (R 4.2.0)
 shiny            1.7.4     2022-12-15 [1] RSPM (R 4.2.0)
 shinyjs          2.1.0     2021-12-23 [1] RSPM (R 4.2.0)
 shinystan        2.6.0     2022-03-03 [1] RSPM (R 4.2.0)
 shinythemes      1.2.0     2021-01-25 [1] RSPM (R 4.2.0)
 StanHeaders      2.21.0-7  2020-12-17 [1] RSPM (R 4.2.0)
 stringi          1.7.12    2023-01-11 [1] RSPM (R 4.2.0)
 stringr        * 1.5.0     2022-12-02 [1] RSPM (R 4.2.0)
 svUnit           1.0.6     2021-04-19 [1] RSPM (R 4.2.0)
 tensorA          0.36.2    2020-11-19 [1] RSPM (R 4.2.0)
 threejs          0.3.3     2020-01-21 [1] RSPM (R 4.2.0)
 tibble         * 3.2.1     2023-03-20 [1] RSPM (R 4.2.0)
 tidybayes      * 3.0.4     2023-03-14 [1] RSPM (R 4.2.0)
 tidyr          * 1.3.0     2023-01-24 [1] RSPM (R 4.2.0)
 tidyselect       1.2.0     2022-10-10 [1] RSPM (R 4.2.0)
 tidyverse      * 2.0.0     2023-02-22 [1] RSPM (R 4.2.0)
 timechange       0.2.0     2023-01-11 [1] RSPM (R 4.2.0)
 tzdb             0.3.0     2022-03-28 [1] RSPM (R 4.2.0)
 utf8             1.2.3     2023-01-31 [1] RSPM (R 4.2.0)
 vctrs            0.6.2     2023-04-19 [1] RSPM (R 4.2.0)
 vroom            1.6.1     2023-01-22 [1] RSPM (R 4.2.0)
 withr            2.5.0     2022-03-03 [1] RSPM (R 4.2.0)
 xfun             0.38      2023-03-24 [1] RSPM (R 4.2.0)
 xtable           1.8-4     2019-04-21 [1] RSPM (R 4.2.0)
 xts              0.13.1    2023-04-16 [1] RSPM (R 4.2.0)
 yaml             2.3.7     2023-01-23 [1] RSPM (R 4.2.0)
 zoo              1.8-12    2023-04-13 [1] RSPM (R 4.2.0)

 [1] /usr/local/lib/R/site-library
 [2] /usr/local/lib/R/library

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